Copy array notation
The first entry is the copiand, the second the iterator, after which the first nonzero entry is the exhauster. This is based on Copy Notation Linear arrays Rules #=aa#^#a (inverse of rule 2) #=Ab(a) where A is defined as: ##A0(a)=aa#^#a=aa##...##a with a'' #s in Copy notation. ##An+1(x)=AnAn(x)(x). Thus, 2-entry CAN is NOT primitive recursive! #<@,0>=<@> #=a #=,c-1@> if b>0 and c>0 #=,d-1#> The symbol for is b¶a, when b is a number. Zeroes can be omitted as in, <99,88,,,,,,,,,,,,,,,,,,,,,,,,,,,,,9999999> as we see in dimensional arrays. Dimensional arrays Rules #In general, =>. ("," is same as 0), if c≥d≥e≥...≥g≥h. Note how ¶ was used as a shorthand for writing out a b^c array. # is a b^c hypercube of ''a, but the last a is replaced with , iff it wasn't replaced. # is a b^c hypercube of a'', but the last a is replaced with , if d>0. # is a b^c hypercube of ''a, but the last a is replaced with #Multi-entry arrays inside brackets are same as BEAF parens, except the & is replaced with ¶ and {} with <>. For example, <3,3,111>=(3¶3)¶3=<3,3,3,...(3¶3 3s)...3,3>. Higher sublegion arrays I'll define ba¶c=a^a^...(b times)...^a^a¶c. To well-define things, @]e>={b,e,@}¶a, continue with pentation(, with φ(2,0) entries; sorry SbS), hexation(, with φ(3,0) entries),... {a,b,1,2}¶c=a{a{...(b "a"s)...{a{a}a}...a}a}a¶c. Then, define tetrexpandal arrays ({a,b,2,2}¶c), pentexpandal arrays ({a,b,3,2}¶c),..., explodal arrays,..., megotional arrays, and even beyond! (NOTE: These operations are first reduced to tetration, then calculated) #The 1st row is of a entries, #the 2nd is of a2-a, #the 3rd is of a3-a2, #:... a. the ''a''th is of aa-aa-1. Then the planes have size b, b2-b,...,bb-bb-1, realms b, b2,...,bb, etc., with dimension aa,2. The tetrational spaces have then b, , >... dimensions. Continue with pentational, hexational,..., and all values are c (except the values outside, being 0). We came to the end of sublegion CAN, and we start the... L1, L2, L3,... arrays We will use a¶↓b=b¶b¶b¶b¶b¶...(a×)...¶b¶b¶b¶b. *Define =. The multi-entry / array is as in BEAF, but {} replaced by angle brackets. <@/0>=<@> *Define =, =. *Then, define =, =, etc. *Then, define =. *Similarly for , ,..., , etc. *Then define = *Similarly for , ,... **Shorthand for /////...(x /s)...///// is /x. (This is a needed notion to go farther than ///...///, since / isn't a number) *So, /c>=cb>. *Continue with //+n, //n, //nm,... **We use then BEAF of /s, to get even farther. *Define = This is only used iff n=cb>. *Continue with /...c>, over which we generalize. *Define b,c=an)a>. Note that 1 is used due to the terminal value being 0. *Continue with L2, L3, L4, L5,..., L^2, L^3,..., L^L,L^L×L,... L arrays (LATEST V.1.01) *We left at L^^L, but we continue using {L,L,n}. We define {L,L+n(1)2}, and even dimensional or tetrational. The impasse has no notation, but we can start using ← for notation, defining L←1=L^^L&L; L←n=L←(n-1)^^L←(n-1)&L←(n-1). *Then we can add more: L←1←1 is the 1st fixed point of x→(L←x), we use ←←, ←←←,... (←2, ←3...) as fixed points of ←, ←←. *This has a supremum of ←←←..., which we define as ←L. *Then define L←←L as ←L←...L. *Continue with 3, 4, 5 superscripted ←: from this we can define a shorthand: x←. *Continue with x=L, and if x=←, then it's 1st fixed point of x|->L x←L. *Then we BEAF the left-arrows. *Continue with subscripts in same manner. We didn't use L-subscripts, which will be the next step. *Lx is the xth fixed point of \(n\mapsto L\,\,_n\!\leftarrow L\). (don't confuse it: this is why the extra space) *Continuing by this manner, we get xL as the xth fixed point of \(n\mapsto L_n\). *By ...LL, we got to the total impasse, ultimate limit of CAN...did we? The \ operator No we didn't. *L\n is the nth fixed point of the ...LL notation. *L\+\n is the nth fixed point of the L\ notation. *Continue and we reach L(\L)L and so on. *The notation's supremum is L\\L, then L\\\L,... *L\nL is L\\\...(n \s)...\\\L *Continue in the exact same manner as left-arrow notation, but using \ instead of ←. Growth rate: Before the notation So here is the comparison. I use spaces because of wikia formatting. Linear Dimensional/Nested Now, these are formats of the separator in where # is the separator Sublegion Now, these will use ¶ Category:Notations Category:Well-defined Category:Semi-naive extensions